Image Analysis for Volumetric Industrial Inspection and Interaction
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1 Image Analysis for Volumetric Industrial Inspection and Interaction Rasmus Larsen Mads F. Hansen, Hildur Olafsdottir, Thomas H. Mosbech, Lasse F. Laursen DTU Informatics
2 (CT) reconstruction from few projections Interpolation of low resolution images The Virtual Knife - interacting with 3D volumes with fast visual feedback
3 Motivation - Industrial application of CT Fast, inexpensive scanning using a minimum of projections Scanning pig backs Quality assesment (lean meat, fat %) Robot control rind trimming Visualization product development
4 Ray Attenuation Model Discretise line integral of attenuation coefficients The measured attenuation b j is a weighted sum of pixels traversed by the ray Pixel weights - length of ray s path through it b j = a 1 x (1,2) + a 2 x (1,3) + a 3 x (2,3) + a 4 x (2,4) + a 5 x (3,4) Resulting linear system b Ax x (1,1) x (1,2) x (1,3) x (1,4) x (2,1) x (2,2) x (2,3) x (2,4) x (3,1) x (3,2) x (3,3) x (3,4)
5 Fan beam - Image Acquisition 100 fan-beam projections of 360 rays each Measured on arc-shaped detector array opposite source
6 Filtered Back Projection Does well with full angle data 360x360 image from 360 source positions with 909 rays each
7 Filtered Back Projection Not so good with sparse data fewer source positions 30 source positions, 909 rays each
8 Reformulate Probabilistic Approach Bayes theorem The loglikelihood of the data log p( x b) p( x) p( b x) 2 p( b x) b Ax / 2 c Pig CTs are piece-wise constant prior should penalise intensity roughness allow for jumps on edges 2
9 Prior Formulation Penalise large gradients - finite difference operators: L 1 x, L 2 x log 2 2 T c T T 1 L x L x x L L L L x c 1 p( x) But only at a sparse set of edge points 1 T T 1 T 1 xθ p θ x L D L L D L x log p θ c log p D diag ( 1,, N ) (, ) j ~ 0
10 Reconstruction Choose (x, ) to maximise the posterior probability Conjugated Gradient Least Squares Two-step algorithm. Fix, minimise wrt. X Fix x, minimise wrt. Q Works also for cone beam CT.... Now some images!
11 Results Only few two-step-sweeps needed x360 image from 100 fans of 540 rays The prior acts as wanted
12 Results 50 fans of 540 rays
13 Results 40 fans of 540 rays
14 Results 30 fans of 540 rays
15 Results 20 fans of 540 rays (pushing it)
16 Results 10 fans of 540 rays (really pushing it)
17 Motivation - Interpolation In-plane resolution higher than through plane Also to compensate for motion Through-plane: 8mm slice thickness + 1mm gap In-plane: 1mm
18 Interpolation Linear interpolation: Weighted average of neighboring voxels z-axis z-axis Original Linear
19 Improved correspondence-based interpolation Adjacent slices: I A, I B Register I A to I B and vice versa, obtain displacements w AB, w BA z-position of desired interpolated value, I gives the distance ratio, α = (z I -z B )/(z A -z B ) Determine intensities at z I between I A and I B : Penney et al. Registration-based Interpolation, IEEE-TMI 2004 Frakes et al. A New method for Registration-based Interpolation, IEEE-TMI 2008 Olafdottir et al. Improving Image Registration using Correspondence Interpolation, ISBI 2011
20 Formulation with respect to Linear Interpolation Improved Correspondence-based interpolation Linear Interpolation Approximation: Assuming smoothness and dense correspondence field
21 Cardiac example Original Original z-gradient with linear interpolation Correspondence-based interp Correspondence-based z-gradient with correspondence-based
22 Gradients of deformed targets, cardiac MRI Linear Spline Correspondence
23 Atlas Building Linear Correspondence
24 Interaction Phantom Omni
25 Interaction by proxy Plane estimation on surface Constant re-evaluation Real-time Performance
26 Linear Interpolation
27 Registration Based Interpolation
28 Motion compensation
29 Conclusion We have demonstrated that Iterative reconstruction using smart priors allows for Image reconstruction using fewer projections at low resolution, and Motion compensation
30 Acknowledgements The Danish Council for Independent Research The Danish Council for Strategic Research Danish Meat Research Institute Aarhus University Hospital, Skejby
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